The hybrid architecture from Sec:scheme allows for the generation of a (2+1)D cluster state suitable for performing scalable quantum computation.
For completeness, we present a possible scheme for the fault-tolerant implementation of logical algorithms on such a state.
The scheme we choose and review here is based on lattice surgery for the surface code [1]}, [2]}, [3]}, [4]}, particularly its measurement-based version [5]}, [6]}.
Note that while the schemes considered here have lower overheads [3]} than other surface-code-based approaches, they are heuristics, since finding an optimal implementation is NP-hard [8]}.
This section overviews the fundamental components needed to perform fault-tolerant quantum information processing in our architecture.
We use as examples codes of distance 5, the distance being the weight of the smallest representative of a logical operator.
Inner (GKP) code states will be denoted by a subscript, whereas outer (RHG) code states will have a bar. The reader may refer to fig:pcell for clarity on the terms primal, dual, rough, and smooth. We use the data and ancilla qubit nomenclature from the foliated picture [9]} where useful; see Fig:patchdefinition for more details.
While much of this section is meant as a review for a photonics audience and might be familiar to an expert in fault tolerance, it does highlight specific points about how these logical operations could be implemented in our architecture.
